The crossing number of a graph is the minimum number of edge crossings in any drawing of in the plane. In the pairwise crossing number, we minimize the number of pairs of edges that cross.

Obviously we have .

The problem was first posed by Pach and Tóth in~[PT], who first spotted the possibility that the pairwise crossing number might be different from the crossing number. They proved for graphs with pairwise crossing number , which was later improved by Valtr~[V05] to and by Tóth~[T08] to .